Seismology

Part IX:

Seismometery

Examples of early attempts to record ground motion.

Definitions:

Seismometer:Transducer from ground motion to something else (usually voltage). Also called geophone.

Seismograph: The recording device. Also called Data Logger.

Seismogram: The record of ground motion.

Seismoscope: (obsolete) Record of ground motion with no time signal (e.g., pendulum in sand)

"Classic" devices date back to the late 1800's. Modern (digital) devices have been evolving for the past several decades. A currently used seismometer (CMG-3T) is shown to the right.

The goal of any device is to provide a high fidelity record of ground motion, to the extent that an investigator and recreate the actual motion from the record.

The Mass on a Spring (Inertial Pendulum)

The simplest kind of transducer is a mass suspended from a casing by a spring. The casing will move with the ground, and the spring provides some kind of decoupling. To understand what happens, we consider Hooke's law:

F = -ky(t)where y(t) is the displacement of the mass from its equilibrium position. Thus

kyym −=&&

describes the unforced (homogeneous) and undamped oscillations of the mass. We recognize this immediately as a simple wave equation with the solution:

€

y = e iωot

where

€

ωo = k /m

Generally, we will concern ourselves with an oscillator driven by a ground motion U(t), and with a damping force proportional to the velocity of the mass, and modify the above to be: ( ) yDkyUym &&&&& −−=+

Uyyyym

Dyy oo

&&&&&&&& −=++=++ 22 2 ωγω

€

γ= D

2m

or

where

If we amplify the ground motion by a factor G, then

UGxxx o&&&&& −=++ 22 ωγ

We consider, without loss of generality, the response to a sinusoidal input:

€

U = e−iωt

and consider a solution of the form X(ω)e-iωt to get

€

X(ω) −ω2 − i2γω +ωo2

( ) =Gω2

€

X(ω ) =−Gω 2

ω 2 −ωo2 + i2γω

=Gω 2

ω 2 −ωo2

( )2+ 4γ 2ω 2

e iφ

€

φ=−tan−1 2γω

ω2 −ωo2

Note that the response is a maximum when ω = ωo (it blows up with no damping). This is called the resonance or characteristic frequency of the system.

If γ << 1, the seismometer will "ring" about ωo. If γ > 1, the response x(t) will not oscillate at all but gradually return to zero. If γ = 1, the oscillations are critically damped.

Note that if ω << ωo, then

€

X(ω ) =Gω 2

−ωo2

( )2∝ω 2

ωo2

€

φ ≈−tan−1 2γω

−ω02

⎛

⎝ ⎜

⎞

⎠ ⎟= −tan−1 0( ) = 0

which means that acceleration is transduced with no phase shift. Thus, accelerometers are designed with very high resonant frequencies.

If ω >> ωo, then

€

X(ω ) =Gω 2

ω 2( )

2= const.

€

φ ≈−tan−1 2γ

ω

⎛

⎝ ⎜

⎞

⎠ ⎟→ π

and so the sensor is directly proportional to ground displacement (the phase shift means that the mass motion is in the opposite direction of U).

The instrument thus responds best to motion with periods near the natural period, and the width of the response will depend on the damping.

Electromagnetic Instruments

Electromagnetic transducers operate by the relative motion of a magnet and a coil, which produces a current in the coil. The effect is generally to modify the above equations with an additional ω term, because the signal generated will be proportional to the velocity of the mass.

Examples:

WWSSN: Sensor, galvanometer, photo paperGeoscope/GSN: Force feedback sensors; digitalIDA:Force feedback gravity meter; digital

NORSAR

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US National Seismic Network

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